The basic idea of the proof in this case, is to note the operator system structure satisfies \(\mathfrak {S}_0 = \mathrm {span}\,\lbrace U_i^* U_j, I \rbrace _{i \ne j}=W^*\mathfrak {A}_\Delta W\)  for some unitary \(W\) . Since \(W^*\mathfrak {A}_\Delta W \cong \mathbb {C}^d\)  has a separating vector and hence Theorem REF  applies. We note that the operator system \(\mathfrak {S}_0\)  was studied in a similar context in [1]}.
